Sets of rigged paths with Virasoro characters

Abstract

Let {M (p,p′) r,s }1≤r≤p−1,1≤s≤p′−1 be the irreducible Virasoro modules in the (p,p′)-minimal series. In our previous paper, we have constructed a monomial basis of ⊕ p−1 r=1 M (p,p′) r,s in the case 1<p′/p<2. By ‘monomials’ we mean vectors of the form \(\phi^{(r_{L},r_{L-1})}_{-n_{L}}\cdots\phi^{(r_{1},r_{0})}_{-n_{1}}{|r_{0},s\rangle }\) , where φ (r′,r)−n :M (p,p′) r,s →M (p,p′) r′,s are the Fourier components of the (2,1)-primary field and |r 0,s〉 is the highest weight vector of \(M^{(p,p')}_{r_{0},s}\) . In this article, we introduce for all p<p′ with p≥3 and s=1 a subset of such monomials as a conjectural basis of ⊕ p−1 r=1 M (p,p′) r,1 . We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.